We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion.This provides a short proof of the recent result due to P. Carr et al that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion is replaced by a stable subordinator.
"A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion." Electron. J. Probab. 14 1532 - 1540, 2009. https://doi.org/10.1214/EJP.v14-674